How to find the maximum profit for -x^2 + 1030x -30,000

Answer 1

Or use the characteristics of a quadratic.

Maximum occurs at -b / 2a.

In this case, -1030 / -2, or 515

Now substitute this value for x and determine maximum.

Answer 2

Find the first derivative of P(x) = -x² + 1030x -30,000: P'(x) = -2x + 1030. This slope of the Profit function will be zero at a maximum or minimum. Second derivative P"(x) = -2, so the slope P' is decreasing and goes from positive to negative at 2x = 1030. So maximum profit is P(515) = 235,225.